Take some compact space, for simplicity take the interval $[0,1]$.
Let $f_n:[0,1] \to [0,1]$ be measure preserving, i.e. $\mu (f_n^{-1}(A))=\mu(A)$ for all lebesgue measurable $A \subset [0,1]$.
The question is under what kinds of convergence $f_n \to f$ do we also ensure that $f$ is measure preserving?
I think I can show that for $f_n \to f$ in $L^2$ this holds. I'm now wondering about weak convergence $f_n \rightharpoonup f$ in $L^2$? Anyone know the answer (and maybe a proof/counterexample)?
Let $f_n(x) = \{nx\}$ where $\{x\}$ denotes the fractional part of $x$. Then $f_n \rightharpoonup \tfrac12$ in $L^2$.